The FoodCourt

Customer Influx as a Poisson Process

A data collector’s account for the influx of customers from respective customer group size allocations presented composition data for a week between 12pm to 2pm. Corresponding group sizes were 1 - 8, and aside from those that wanted to dine in, there were also individuals that simply came into the shop for takeout/takeaway.

The composition of the customer base was noted. Similarly, the total number of customers that entered for the week per day, was noted. However, in the case of the simulation, a fixed number of customers was not set at the beginning to enter the shop and so forth. Instead, what was done was an emulation of 2 homogeneous Poisson process - one for each section of the peak hour (2/3 of the customers entering between the 60min period between 1215 and 1315, and the remaining 1/3 in the final 45min until 1400).

On Monday, 248 customers came in. On Tuesday, that was 206, and so on and so forth. The Poisson process, with the appropriate rate, can be used to approximately simulate a process whereby customers enter the foodcourt such that the total number of customers who visited the shop, is similar to that which was observed.

As stated in the data collection report, it was not entirely clear if all customers who wanted to come in, truly were able to come in, and purchase their desired products. There was an understanding that generally all customers that had entered the food court, were able to eventually get their food. In other words, there is no clear metric for the customers who were discouraged from entering the food court due to issues such as crowding.

A bold simplification | Difference between customers and customer groups

To simplify the process, a rate of 2.8 was estimated to be roughly appropriate for this simulation. This rate is not the true rate of customer groups coming. However, by virtue of the uncertainty involved in this case, it would seem reasonable to simply take the multinomial distribution of the group sizes at face value, and work backwards later to ascertain the rate of actual customer group influx.

If one were to take a genuine attempt at ascertaining the customer GROUP influx (in place of generic customer influx), one would incorporate a multinomial distribution into this simulation and attempt to work backwards.

The addendum

real<- function(cust_group_no){
  if (cust_group_no==0){
    return(0)
  }
  summie <- 0
  for (i in 1:cust_group_no){
    cv<-rmultinom(1,1,c(28,26,46,18,5,1,1,1,2))
    cvlist<-to_vec(for(i in 3:9) cv[i]*(i-1))
    summie<-summie+sum(cv[1:2],cvlist)
  }
  return(summie)
}

In the code, upon review, it has become clear that the applied rate of Customer generation, not to be confused with the individual customers, is meant to be approximately 1.2 - 1.5 as the base rate, followed by of course, the necessary, adjusted Customer influx rate.

As it is apparent above, the total number of customers generated here is a true reflection of the joint operation between the Poisson generation of the Customer group AND the multinomial distribution of the group size.

Table array

A fixed array of 48 spaces. Initialized with values of 0, to represent unoccupied seats and they are updated with the occupying Customer’s eating time. Interested Customers do not occupy the table unless 2 conditions are fulfilled.

Firstly, the Waiting List for the Tables have to be empty.

Secondly, the tables have to have enough space for ALL the customers in the group to occupy.

Simple array in construction.